Chapter 1 Paraconsistency: Introduction Koji Tanaka, Francesco Berto, Edwin Mares, and Francesco Paoli 1.1 Logic It is a natural view that our intellectual activities should not result in positing contradictory theories or claims: we ought to keep our theories and claims as consistent as possible. The rationale for this comes from the venerable Law of Non- Contradiction, to be found already in Aristotle’s Metaphysics, and which can be formulated by stating: for any truth-bearer A, it is impossible for both A and :A to be true. Dialetheism, the view that some true truth-bearers have true negations, challenges this orthodoxy.1 If some contradictions can be true, as dialetheists have argued, then it may well be rational to accept and assert them. For example, one may think that the naïve account of truth, based on the unrestricted T -schema: hAi is true 1Dialetheism itself has a venerable tradition in the history of Western philosophy: Heraclitus and other pre-Socratic philosophers were arguably dialetheists, for instance; and so were Hegel and Marx, who placed the obtaining and overcoming (Aufhebung) of contradictions at the core of their ‘dialectical method’. For an introduction to dialetheism, see Berto and Priest (2008). A notable collection of essays on the Law of Non-Contradiction is Priest et al. (2004). K. Tanaka () University of Auckland, Auckland, New Zealand e-mail: [email protected] F. Berto University of Aberdeen, Aberdeen, UK e-mail: [email protected] E. Mares Victoria University of Wellington, Wellington, New Zealand e-mail: [email protected] F. Paoli University of Cagliari, Cagliari, Italy e-mail: [email protected] K. Tanaka et al. (eds.), Paraconsistency: Logic and Applications, Logic, Epistemology, 1 and the Unity of Science 26, DOI 10.1007/978-94-007-4438-7__1, © Springer Science+Business Media Dordrecht 2013 2 K. Tanaka et al. if and only if A, should be accepted on a rational ground because of its virtues of adequacy to the data, simplicity, and explanatory power. However, the account is inconsistent, due to its delivering semantic paradoxes, such as the Liar.2 A dialetheist had better not be a classical logician. Classical logical conse- quence supports the principle often called ex contradictione quodlibet (ECQ): fA; :AgˆB for any A and B. We are licensed by classical logic to infer anything whatsoever when we end up with a contradiction. To use a lively expression, classical logic is explosive: the truth of everything—a view often called trivialism— is classically entailed by the obtaining of a single contradiction; and trivialism is rationally unacceptable if anything is.3 A necessary condition for a logic to be paraconsistent is that its logical consequence relation, ˆ, is not explosive, invalidating ECQ. Although there is no general consensus on a definition of paraconsistent logic among researchers in the area, more often than not this necessary condition is taken to be a sufficient one too. Some logicians,4 on the other hand, have argued that this ‘negative’ constraint should be supplemented by appropriate additional ‘positive’ properties. Be it as it may, since paraconsistent logics do not allow us to infer anything arbitrarily from a contradiction, their treatment of inconsistencies appears more sensible than the one in classical logic. But whereas a dialetheist should go paraconsistent, one does not need to accept that there are true contradictions to adopt a paraconsistent logic.5 Dialetheism is a controversial view and many people find it counterintuitive. But, regardless of whether there are some true contradictions, it may be that in most cases when we find that we hold inconsistent beliefs or make inconsistent claims, we should revise them to be consistent.6 Whether or not there are no true contradictions, inconsistency is pervasive in our rational life. We often find that we have inconsistent beliefs or make inconsistent claims, and we are often subject to inconsistent information. Any philosopher who thinks that we may use a logic to make inferences from, determine commitments of, or otherwise logically examine the contents of people’s beliefs, theories, or stories, should therefore think twice before being committed to explosion. For example, telling someone who has contradictory beliefs that they are committed to believing every proposition would be a very unproductive move in most debates, and do little more than merely pointing out that the person has inconsistent beliefs. Such considerations provide independent motivations for the development of paraconsistent logics: we need subtle, non-classical logical techniques to analyse the features of inconsistent theories and beliefs. 2See Priest (2005), Chap. 7. 3Trivialism finds, however, a recent, brilliant defence in Kabay (2010). 4See Béziau (2000). 5See Berto (2007) Chap. 5 and Priest and Tanaka (2009). 6Even dialetheists accept this. See, for example, Priest (2005) Chap. 8. For paraconsistent belief revision, see Mares (2002)andTanaka (2005). 1 Paraconsistency: Introduction 3 The history of paraconsistent logic has taught us that just taking classical logic and barring ECQ is not sufficient to produce an interesting non-explosive logic. In fact, a number of distinct logical techniques to invalidate ECQ have been proposed. As the interest in paraconsistent logic has grown, different people at different times and places have developed different non-explosive perspectives independently of each other. As a result, the development of paraconsistent logics has somewhat a regional flavour. This book is not a technical survey of the variety of paraconsistent logics7: it aims at illustrating their philosophical motivations, applications, and spin-offs. Since these logics are little known to non-specialists, though, in what follows we briefly summarise the most prominent logical strategies to achieve paraconsistency which feature in, or are presupposed by, the essays in this volume. 1.1.1 Discursive Logic The first formal paraconsistent logic was developed in 1948 by the Polish logician Jaskowski,´ in the form of discussive (or discursive) logic.8 Jaskowski’s´ approach addressed situations involving distinct cognitive agents each putting forth her own beliefs, opinions, or reports on some event or other. Each participant’s opinions may be self-consistent. However, the resultant discourse or set of data as a whole, taken as the sum of the assertions put forward by the participants, may be inconsistent. Jaskowski´ formalised this idea by modelling the inconsistent dialogical situation in a modal logic. For simplicity, Jaskowski´ chose S5. We think of each participant’s belief set (or set of opinions, assertions, etc.) as the set of sentences true at a world in a S5 model M . Thus, a sentence A asserted by a participant in a discourse is interpreted as “It is possible that A”(ÞA). That is, a sentence A of discussive logic can be translated into a sentence ÞA of S5. Then A holds in a discourse iff A is true at some world in M .SinceA may hold in one world but not in another, both A and :A may hold in a discourse. In this volume, however, Marek Nasieniewski and Andrzej Pietruszczak show how Jaskowksi’s´ discussive logic can also be expressed via normal and regular modal logics weaker than S5 in their essay On Modal Logics Defining Ja´skowski’s D2-Consequence. 1.1.2 Preservationism In a discursive logic, a consequence relation can be thought of as defined over maxi- mally consistent subsets of the premises. Given a set of premises, we can measure its degree of (in)consistency in terms of the number of its maximally consistent subsets. 7For surveys, besides Priest and Tanaka (2009), see Priest (2002)andBrown (2002). 8See Jaskowski´ (1948). 4 K. Tanaka et al. For example, the level of fp; qg is 1 since the maximally consistent subset is the set itself. The level of fp; :pg is 2 since there are two maximally consistent subsets. If we define a consequence relation over some maximally consistent subset, then the relation can be thought of as preserving the level of consistent fragments. This is the approach which has come to be called preservationism. It was first developed by the Canadian logicians Ray Jennings and Peter Schotch.9 In this volume, Bryson Brown’s essay Consequence as Preservation: Some Refinements moves within this tradition, but proposes a more general view of the features a logical consequence relation can be seen as preserving. 1.1.3 Adaptive Logics One may think that we should treat a sentence or a theory as consistently as possible. However, once we encounter a contradiction in reasoning, we should adapt to the situation. Adaptive logics, developed by Diderik Batens and his collaborators in Belgium, are logics that ‘adapt’ themselves to the (in)consistency of a set of premises available at the time of application of inference rules. As new information becomes available expanding the premise set, consequences inferred previously may have to be withdrawn. However, as our reasoning proceeds from a premise set, we may encounter a situation where we infer a consequence provided that no abnormality, in particular no contradiction, obtains at some stage of the reasoning process. If we are forced to infer a contradiction at a later stage, our reasoning has to adapt itself so that an application of the previously used inference rules is withdrawn. Adaptive logics model the dynamics of our reasoning as it may encounter contradictions in its temporal development.10 In this volume, Diderik Batens’ essay New Arguments for Adaptive Logics presents four new arguments vindicating the utility of the adaptive approach. 1.1.4 Logics of Formal Inconsistency The approaches to paraconsistency we have referred to so far retain as much classical machinery as possible (many paraconsistent logicians believe that the full inferential power of classical logic ought to be retained as much as possible, insofar as we find ourselves in consistent contexts).